# Connections Between Metric Coefficients, Basis- And Unit Vectors

### Abstract

AbstractThis work gathers in one place what is pertinent about the connections between metric coefficients, basis- and unit vectors in a four-dimensional relativistic manifold. Some of this material can be found scattered elsewhere; its collection into one place reveals connections that either are not known or are obscure, for example, that the metric coefficients are not all independent of each other. It at least should serve as a useful tutorial for those who are not thoroughly familiar with this material.

### References

[1] Misner C. W., Thorne K. S. and Wheeler J. A., Gravitation. Princeton U. Press,

2017.

[2] Hartle J. B., Gravity, An Introduction to Einstein's General Relativity, Addison-

Wesley, 2002.

[3] M. P. Hobson, G. Efstathiou, and A. N. Lasenby, General Relativity, An

Introduction for Physicists, Cambridge University Press, 2006.

[4] Schutz B. F., A First Course In General Relativity, Cambridge U. Press, 1985.

[5] Pauli W., Theory of Relativity, Pergamon Press, 1958.

[6] Bergmann P. G., Introduction To The Theory of Relativity, Dover Publications, 1976.

[7] De Wolf D.A. , Basis Vectors in Relativity, Eur. J. Phys. Ed., 12, 2021.

[8] See ref. 1, §9.7ref. 2, §2.1, or https://en.wikipedia.org/wiki/Manifold

[9] See ref. 8 and in particular ref. 3, §2.1

[10] https://en.wikipedia.org/wiki/Curvilinear_coordinates, https://www.physicsforums.com/threads/general-relativity-and-curvilinear-coordinates.818066/

[11] Covariant vectors are indicated with subscripts and contravariant ones with super-

scripts. See also ref. 3, §3.4 and §3.5, or https://www.seas.upenn.edu/~amyers/

DualBasis.pdf.

[12] See ref. 2. §20.2.

[13] https://en.wikipedia.org/wiki/Kerr_metric.

[14] Ref. [2], section 8.4

2017.

[2] Hartle J. B., Gravity, An Introduction to Einstein's General Relativity, Addison-

Wesley, 2002.

[3] M. P. Hobson, G. Efstathiou, and A. N. Lasenby, General Relativity, An

Introduction for Physicists, Cambridge University Press, 2006.

[4] Schutz B. F., A First Course In General Relativity, Cambridge U. Press, 1985.

[5] Pauli W., Theory of Relativity, Pergamon Press, 1958.

[6] Bergmann P. G., Introduction To The Theory of Relativity, Dover Publications, 1976.

[7] De Wolf D.A. , Basis Vectors in Relativity, Eur. J. Phys. Ed., 12, 2021.

[8] See ref. 1, §9.7ref. 2, §2.1, or https://en.wikipedia.org/wiki/Manifold

[9] See ref. 8 and in particular ref. 3, §2.1

[10] https://en.wikipedia.org/wiki/Curvilinear_coordinates, https://www.physicsforums.com/threads/general-relativity-and-curvilinear-coordinates.818066/

[11] Covariant vectors are indicated with subscripts and contravariant ones with super-

scripts. See also ref. 3, §3.4 and §3.5, or https://www.seas.upenn.edu/~amyers/

DualBasis.pdf.

[12] See ref. 2. §20.2.

[13] https://en.wikipedia.org/wiki/Kerr_metric.

[14] Ref. [2], section 8.4

Published

2023-08-27

How to Cite

DE WOLF, David A.
Connections Between Metric Coefficients, Basis- And Unit Vectors.

**European Journal of Physics Education**, [S.l.], v. 14, n. 2, p. 33-44, aug. 2023. ISSN 1309-7202. Available at: <https://www.eu-journal.org/index.php/EJPE/article/view/371>. Date accessed: 13 july 2024.
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Section

Classroom Physics

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